5-4 Completing the Square LESSON You can use the square root property to solve some quadratic equations. Square Root Property To solve x 2 a, take the...
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Name
Date
Class
Reteach
LESSON
5-4
Completing the Square
You can use the square root property to solve some quadratic equations. Square Root Property 2 To solve x a, take the square root of both sides of the equation.
x2 a
Remember:
a x 2
2 2 2 4, and 2 4.
x a
2 Solve 4x 5 43. 2 4x 48
The coefficient of x 2 should be 1 to use the square root property.
You can use a process called completing the square to rewrite a quadratic of the form x 2 bx as a perfect square trinomial. To complete the square of x 2 bx, b 2. add __ 2
Think: Multiply the coefficient of x by 1 . Then square it. __ 2
x __2b
2 b x bx __ 2
2
2
Complete the square: x 2 8x ?. Identify b, the coefficient of x : b 8.
Step 1
Step 2 Step 3 Step 4
8 __b2 ___ 2
b 2: Find __ 2 b 2: Add __ 2
2
x 2 8x 16
Factor:
x 2 8x 16 x 4 2
2
4 2 16
Check: x 4 2 x 4 x 4
b as a factor. Use __ 2
x 2 8x 16 ✓ Complete each square and factor. 2 5. x 9x ?
7RITE EACH FUNCTION IN VERTEX FORM AND IDENTIFY ITS VERTEX
p^^^
X X
^ Êr {£ÊÊ ÚÚÚÚ
X X B TO EACH SIDE OF THE EQUATION A !DD ??
D D
ÊÊÊTXÊÊÈÊEÊÓÊ
3OLVE EACH EQUATION BY COMPLETING THE SQUARE
???
3OLVE EACH EQUATION BY COMPLETING THE SQUARE
X r
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X X
S D
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X X
X X
??
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X X
X X
XÊÊÇÊÊÎÊrÓÊÊ
#OMPLETE THE SQUARE FOR EACH EXPRESSION 7RITE THE RESULTING EXPRESSION AS A BINOMIAL SQUARED
4O COMPLETE THE SQUARE OF X BX ADD ?? B TO THE EXPRESSION 7RITE THE TERM NEEDED TO COMPLETE THE SQUARE FOR EACH EXPRESSION
^
^ XÊÊÓÊrÈÊÊ
X X
X
^ X r
G TX E ÊÊÊÊTX ÓÊEÊÓÊ ÎÆÊÊÊTÓ]ÊÎÊEÊ F S X D X X
H TX E ÊÊÎÊÊTX £ÊEÊÓÊ £nÆÊÊÊT£]Ê£nÊEÊ
F TX E ÊÊÓÊÊTXÊÊ{ÊEÊÓÊÊÎÈÆÊÊÊT{]ÊÎÈÊEÊ
3OLVE
3OLVE
.ATHAN MADE A TRIANGULAR PENNANT FOR THE BAND BOOSTER CLUB 4HE AREA OF THE PENNANT IS SQUARE FEET 4HE BASE OF THE PENNANT IS FEET SHORTER THAN THE HEIGHT
2ALPH AND %DIE EACH SOLVED THE EQUATION S X D 2ALPH SAYS THE CORRECT ANSWER IS X %DIE SAYS THE CORRECT ANSWER IS X 7HO IS CORRECT (OW DO YOU KNOW
A 7HAT ARE THE LENGTHS OF THE BASE AND HEIGHT OF THE PENNANT
4HEY ARE BOTH CORRECT 0OSSIBLE ANSWER ! QUADRATIC CAN HAVE TWO POSSIBLE SOLUTIONS X SO X
>ÃiÊÊnÊvÌ]Ê i} ÌÊÊÓäÊvÌ
B 7HAT ARE THE DIMENSIONS OF THE PENNANT IF THE BASE IS ONLY FEET SHORTER THAN THE HEIGHT
4HE ANCIENT 'REEKS CONSTRUCTED RECTANGLES CALLED GOLDEN RECTANGLES BECAUSE THEY WERE THOUGHT TO BE PLEASING TO THE EYE ! RECTANGLE IS CONSIDERED GOLDEN IF THE DIMENSIONS OF THE RECTANGLE ARE IN A CERTAIN RATIO W ?? WL L ????? L 4HE RATIO ?? WL IS CALLED THE GOLDEN RATIO ! GOLDEN RECTANGLE
9OU CAN USE A PROCESS CALLED COMPLETING THE SQUARE TO REWRITE A QUADRATIC OF THE FORM X BX AS A PERFECT SQUARE TRINOMIAL
4HINK -ULTIPLY THE COEFFICIENT OF X BY 4HEN SQUARE IT ??
4O COMPLETE THE SQUARE OF X BX B ADD ??
S D
S D S
X BX ?? B X ?? B
D
WITH LENGTH L AND WIDTH W HAS THE PROPERTY THAT IF IT IS JOINED TO A SQUARE OF SIDE LENGTH L TO FORM A LARGER RECTANGLE THE LENGTH TO WIDTH RATIO OF THE LARGER RECTANGLE IS THE SAME AS THAT OF THE ORIGINAL RECTANGLE
#OMPLETE THE SQUARE X X 3TEP
)DENTIFY B THE COEFFICIENT OF X B
3TEP
&IND ?? B !DD ?? B
S D S D
3TEP 3TEP
&ACTOR
S D S D
W
X X
A #LEAR THE EQUATION OF FRACTIONS AND COLLECT ALL THE TERMS THAT CONTAIN VARIABLES ON THE LEFT SIDE OF THE EQUATION
X X S X D
Ó
S D
X X
S
X
??
D
S D
???
TX
TX E
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B 7HAT DECIMAL VALUE DO THESE RATIOS APPROXIMATE / iÊ}`iÊÀ>Ì]Ê£°È£n AS THE LIST IS CONTINUED #ONSIDER THE CONTINUED FRACTION ????????????? ?????????? ?????? -AKE A TABLE OF DECIMAL VALUES FOR THIS FRACTION WHEN FRACTION IS USED THEN THEN AND SO ON 2OUND THE VALUES TO THE NEAREST THOUSANDTH 7HAT VALUE DO THE FRACTIONS SEEM TO APPROACH
*UST AS SOME NUMBERS ARE PERFECT SQUARES SOME QUADRATIC EXPRESSIONS ARE PERFECT SQUARES 0ERFECT SQUARE S X D X X 9OU CAN MODEL THIS EXPRESSION WITH THE AREA OF ALGEBRA TILES 9OU CAN USE THREE TYPES OF TILES
3EAN THINKS THE FLARE SHOULD REACH AT LEAST METERS TO BE SEEN FROM THE SHORE 4HEY WANT TO KNOW HOW LONG THE FLARE WILL TAKE TO REACH THIS HEIGHT
B 3IMPLIFY THE FUNCTION SO YOU CAN COMPLETE THE SQUARE C 3OLVE THE EQUATION BY COMPLETING THE SQUARE
H TT E ÊÊxÊÊTTÊÊÎÊ Ê ÊÊ{x / iÊVÃÌ>ÌÊÌiÀÆÊ{xÊiÌiÀÃ
! 4HE FLARE WILL REACH M IN S
" S
" 4HE FLARE WILL REACH M IN S
# S
# 4HE FLARE WILL REACH M IN S
$ S
$ 4HE FLARE WILL REACH M IN S
X X
X
X
!DD
S D
0ERFECT 3QUARE
?? B
??
S D
S
X ?? X BX ??? B B
D
X X S X D
!NSWER EACH QUESTION #IRCLE THE EXPRESSIONS THAT ARE PERFECT SQUARES
S X D
X X
S X D
A 7HAT WOULD YOU ADD TO THE EXPRESSION X X TO MAKE IT A PERFECT SQUARE B 7RITE THIS EXPRESSION AS A PERFECT SQUARE
X X
X X
X
{ TXÊÊÓÊEÊÓÊ
A 5SE ALGEBRA TILES TO DRAW A MODEL FOR THE EXPRESSION X X B 7RITE THIS EXPRESSION AS A PERFECT SQUARE 7HAT ARE THE SIDES OF THE SQUARE CREATED IN THE MODEL
#HOOSE THE LETTER FOR THE BEST ANSWER 5SE THE VERTEX FORM OF THE FUNCTION TO DETERMINE HOW LONG AFTER FIRING THE FLARE IT WILL REACH ITS MAXIMUM HEIGHT
X
3EAN WANTS TO KNOW HOW HIGH THE FLARE WILL REACH ABOVE THE SURFACE OF THE WATER
B #OMPLETE THE SQUARE USING THE VERTEX FORM OF THE FUNCTION
S X D X X
2ECTANGLE WITH SIDES X AND
E 3EAN THINKS THE FLARE WILL REACH METERS SOONER BUT THEN THE FLARE WILL STAY ABOVE METERS FOR ABOUT SECONDS )S HE CORRECT %XPLAIN
A 7RITE THE FUNCTION IN VERTEX FORM FACTORING SO THE COEFFICIENT OF T IS
D -ASON THINKS THAT THE FLARE WILL REACH METERS IN SECONDS )S HE CORRECT %XPLAIN
ÌÊ}iLÀ>ÊÓ
,i>`}Ê-ÌÀ>Ìi}Þ 5SE A -ODEL
--"
3EAN AND -ASON RUN OUT OF GAS WHILE FISHING FROM THEIR BOAT IN THE BAY 4HEY SET OFF AN EMERGENCY FLARE WITH AN INITIAL VERTICAL VELOCITY OF METERS PER SECOND 4HE HEIGHT OF THE FLARE IN METERS CAN BE MODELED BY H TT E T T WHERE T REPRESENTS THE NUMBER OF SECONDS AFTER LAUNCH
A )N THE &IBONACCI 3EQUENCE OÊ P EACH TERM FROM THE THIRD TERM ON IS THE SUM OF THE PREVIOUS TWO TERMS -AKE A LIST OF VALUES OF THE RATIO OF A TERM AND IT PREDECESSOR
E
T E
E
-EASURE THE LENGTH AND WIDTH OF A CREDIT CARD AND CALCULATE THE RATIO OF THE LENGTH AND WIDTH $OES THIS CLOSELY APPROXIMATE THE GOLDEN RATIO
T
X X
?? B
X X
T E
B SO ?? B
???
Ó
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X X
B SO ?? B ??
Ó
B #OMPLETE THE SQUARE AND SOLVE FOR L IN TERMS OF W )GNORE THE NEGATIVE SOLUTION SINCE L MUST BE A POSITIVE NUMBER 5SE THE RESULT TO FIND BOTH THE L EXACT VALUE OF ?? W AND A DECIMAL APPROXIMATION
